A Numerical Method for Linear ODEs using Non-recursive Haar Connection Coefficients

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1 International Journal of Computational Science and Matematics ISSN Volume, Number 3 (00), pp International Researc Publication House ttp://wwwirpousecom A Numerical Metod for Linear ODEs using Non-recursive Haar Connection Coefficients Monika Garg and Lillie Dewan Electrical Engineering Department, National Institute of ecnology Kuruksetra, Kuruksetra, Haryana, India monika_mittalkkr@rediffmailcom Electrical Engineering Department, National Institute of ecnology Kuruksetra, Kuruksetra, Haryana, India l_dewan@nitkkracin Abstract In tis paper, a novel numerical metod, based on non-recursive Haar connection coefficients, for solving linear ordinary differential equations is presented Based on te pioneering Non-recursive formulation of Haar connection coefficients a generalized numerical metod is presented for solving different types of LODEs using te simple rules of matrix-algebra e simplicity, effectiveness, accurateness and generalized nature of te proposed metod is demonstrated wit te elp of several numerical examples Keywords: Haar wavelet, Haar connection coefficients, Differential equations Introduction Linear Ordinary differential equations (LODEs) describe many of te processes occurring in te fields of pysics, medicine, engineering, nature, economics etc e solutions of tese ODEs ave to be found so as to understand and analyze te dynamics of different processes ese ODEs occur in different variants; omogenous/in-omogenous equations wit constant or variable coefficients Generally, it is tedious to solve LODEs using analytic metods suc as Laplace ransform, cange/variation of variables etc e situation becomes worse wen te order of te differential equations is ig or wen te coefficients are variables e problem is not less complicated in te case of constant coefficients wen multiple iterations ave to be carried out so as to find te complete solution space; suc as

2 430 Monika Garg and Lillie Dewan finding te range of parameter wic gives non-trivial solution satisfying te given boundary conditions as in Sturm Liouville problems [] is is a general secondorder differential equation wic leads to specific LODEs, like Bessel equations and Legendre equations wit variable coefficients, for different values of coefficient functions Also, in te case of in-omogenous differential equations te solution is to be found in two parts; complementary solution to te corresponding omogeneous equation and particular solution for te corresponding forcing function Moreover, even after solving te differential equation te question of computing te values remain Numerical metods provide answer to tese problems [,3] Wavelet based numerical metods are gaining prominence due to nice properties of wavelets like multiresolution and compact support Haar wavelet is te lowest member of Daubecies family of wavelets and is most convenient for computer implementations due to te availability of explicit expression for te Haar scaling and wavelet functions [4] Operational approac, pioneered by Hsiao et al [5], as recently been used for solving LODEs In tis approac, te integro-differential equations are converted into linear matrix-algebraic equations by replacing te matematical operations of integration and differentiation etc by corresponding operational matrices suc as integral operational matrix and connection coefficient matrix Connection coefficient matrix arise wen te product of two functions are to be expanded in terms of Haar bases Hiterto, recursive product matrix and connection coefficient matrix as been reported to be used for te application of solving stiff differential equations and functional differential equations [6,7] However, tese recursive formulations are computationally costlier as iger resolution matrices are to be computed using all matrices at lower resolutions In tis paper, a pioneering non-recursive formulation of Haar connection coefficient matrix is derived based on wic a novel numerical metod is proposed for solving different variants of LODEs in an unified framework and te complete solution is obtained by te single application of te proposed metod For integral operational matrix, te non-recursive formulation derived by Li et al [8] is used e present novel non-recursive formulation [9] pioneered by us, is computationally more efficient and convenient for computer implementations as compared to corresponding recursive formulations reported in te literature so far is paper is organized as follows: Non-recursive formulation of Haar product matrix and connection coefficients is derived in Section In Section, te proposed novel numerical metod for solving LODEs is presented wic is based on te derived non-recursive Haar connection coefficients e proposed metod is applied on several numerical examples in Section 3, followed by te results & discussions and conclusions in te end Haar Wavelet e ortogonal set of Haar wavelets n () t is a group of square waves wit magnitude of ± in certain intervals and zeros elsewere [4] e first function is Haar scaling function () t followed by Haar wavelet function 0 () t as te second function All te

3 A Numerical Metod for Linear ODEs using Non-recursive Haar 43 oter functions are dilations and translations of Haar wavelet function In general, Haar wavelet series is defined as ( t) =, 0 t <,, 0 t < 0 () t = (), t < n () t = ( t l), n = + l, j 0, 0 l < were j & m = j n = m j j j l indicate dilations and translations respectively e resolution m is given by and 0, e symbolic form of te Haar wavelet matrix Hm () t is defined as Hm( t) = [ 0( t) ( t) m ( t)] () e numeric form of Haar wavelet matrix H m is te sampled values of Haar wavelets, arranged as rows For example H8 = Next, non-recursive formulation of different Haar operational matrices required for solving LODEs is derived a Product operational matrix Product operational matrix is defined for te product of two Haar wavelet matrices as Hm() t Hm() t Eac term in te above product is expressed in terms of block pulse functions (BPF) Bm () t and re-arranged as Hm() t Hm() t = Hmdiag( Bm()) t Hm (3) Here, eac block pulse function is defined to be unity in an unit interval of time and zero elsewere and expressed collectively as Bm( t) = [ b0( t) b( t) bm ( t)] were bi ()' t s are individual BPF For details see [9] e product matrix derived in (3) is te non-recursive formulation pioneered by us vis-à-vis corresponding recursive formulation reported in te literature [7] as M / /() / / ( ) m m t Hm m diag b Hm() t Hm() t Mm m() t = diag( b) H diag( H ) (4) m/ m/ m/ m/ a Were [ 0,,, / ] a m and b [ m/, m/+,, m ] It is evident tat te proposed non-recursive formulation in (3) is simple and computationally efficient as compared to recursive formulation in (4)

4 43 Monika Garg and Lillie Dewan b Connection coefficients operational matrix Connection coefficients operational matrix is used to express te operation of product of two Haar wavelet matrices on Haar expansion coefficients of any square integrable function f() t to express it as simple Haar expansion as Hm() t Hm() t c = CHm() t (5) Were C are Haar connection coefficients and c [ 0 ( ) ] = c c c m are Haar expansion coefficients of f() t e function f() t can also be expanded in block pulse functions and Haar wavelet as f() t = cbbm() t = chm() t = chmbm() t (6) were c [ 0 ( ) ] b = cb cb cb m are block pulse function expansion coefficients Using (3)-(5), C is obtained as C = Hmdiag( cb) H m (7) For details one can refer to [9] e Haar connection coefficients proposed in (7) is te non-recursive formulation pioneered by us vis-à-vis corresponding recursive formulation reported in te literature [7] as Hm() t Hm() t Mm m() t cm = Cm mhm() t Cm/ m/ Hm/ m/diag( cb) (8) Cm m diag( cb) H diag( c / / / /) m m bh m m were [ 0,,, / ] ca c c cm and c b [ cm/, cm/+,, cm ] and cm [ c0, c,, cm ] It is evident tat te proposed non-recursive formulation in (7) is simple and computationally efficient as compared to recursive formulation in (8) Proposed numerical metod for ODEs Consider a general linear ODE of order n can be represented as: n n n d y d y d y Px ( ) + Qx ( ) + Rx ( ) + Sxy ( ) = Fx ( ) (9) n n n Were P(), Q(), R() and S () are te coefficient functions and F () is te forcing function of te independent variable wic is x in tis case Let n n d y d y ( 0 ) = a, ( 0 ) = b,, y( 0 n n ) = c are te given initial conditions For simplicity assume x [0 ) is is not a strict restriction as te proposed metod can be easily modified for any x [0 t f ), were t f is te arbitrary final time Since te differentiation of Haar functions lead to impulses, so te igest order differentiation term is expanded in terms of Haar wavelet first Here independent variable is given as x and ence Hm( x ) is used instead of Hm() t e corresponding expansions are as follows:

5 A Numerical Metod for Linear ODEs using Non-recursive Haar 433 n d y Let ch m( x) n = (0) were c = [ c0 c c( m ) ] are coefficients of expansion at te resolution m Integrating (0) once wrt x yields t n t n d y d y = ch m( x) + ( 0 n n ) () 0 0 Using forward operational matrix of integration Q m derived by Lin Wu et al [7] and n d y 0 = a, () is written as n n d y = cq ( ) mhm x+ i Hm( x) () n Were a = i H ( x) is te Haar expansion of initial value a and substituting te given initial value ( ) m i = [ i 0 i i ( m ) ] are te corresponding expansion coefficients Repeating te process again for eqn () and substituting te given initial value n n d y d y ( 0 n ) = b, te value of is obtained as n n d y = ( cq m + i ) QmHm( x) + ihm( x) (3) n Similarly ( ) ( ) y = cqm + i Qm + i + QmHm( x) + inhm( x) (4) Were b= i H ( x) is te Haar expansion of initial value b and m i = [ i0 i i ( m ) ] are te corresponding expansion coefficients Haar expansions of coefficient functions Px ( ), Qx ( ), Rx ( ) and Sx ( ) and forcing function F( x ) are defined as Px ( ) = ph m( x) Qx ( ) = qh m( x) (5) Rx ( ) = rh m( x) Sx ( ) = sh m( x) Were p = [ p0 p p( m ) ], q = [ q0 q q( m ) ], r = [ r0 r r( m ) ] and s = [ s0 s s( m ) ] are te corresponding expansion coefficients Substituting different expansions from (0) (5), (9) becomes ph( xch ) ( x) + qh( x) cq + i H( x) + = sh( x) (6) ( ) m m m m m m Rearranging (6) gives c H ( x) H ( x) p + c Q H ( x) H ( x) q + i H ( x) H ( x) q + = s H ( x) (7) m m m m m m m m

6 434 Monika Garg and Lillie Dewan Using proposed non-recursive connection coefficient formulation from (5), (7) simplifies to c ˆ ( ) ˆ ( ) ˆ phm x + cqmqhm x + i qhm( x) + = shm( x) (8) Equation (8) is a linear matrix algebraic equation wic can be simplified using linear algebra principles as c pˆ + Q qˆ H ( x) = s i qˆ + H ( x) (9) ( ) ( ) ( ) m m m Rigt multiplying wit H m ( x) and using te identity Hm( x) Hm ( x) = I (Identity matrix), (9) can be solved for c as c ( ( ˆ ))( ˆ ˆ ) = s i q + p + Qmq (0) Using te value of c from (0) te value of y is obtained using (4) e simplicity, effectiveness and te capability to tackle all variants of LODEs is demonstrated wit te elp of numerical examples in te next section Numerical Examples e proposed numerical metod is applied to several numerical examples encompassing different variants of LODEs e proposed metod is applied according to te following algoritm: Step Assume te igest order differential as unknown and expand in terms of Haar wavelet Step Integrate repeatedly and substitute te given initial conditions at every integration Expand te constant initial condition in terms of Haar wavelet at intermediate steps Step 3 Expand all te coefficients and forcing function in terms of Haar wavelet Step 4 Substitute all te expansions in ODE and use connection coefficients to expand te product of two functions in terms of simple Haar expansion Step 5 Use te principles of linear algebra to solve for unknown Step 6 Obtain te desired numerical solution of yx ( ) as y_aar and compare it wit te exact solution y_exact Step 7 Obtain te L - norm error using norm{ ( y_aar y_exact ),} % Relative error = 00 norm{ ( y_exact )} Example d y( x) Solve te equation + yx ( ) = sinx+ xcos x, x= [0 ) Given initial dy conditions ( 0) = and y ( 0) = Exact solution reported in te literature is 5 yx ( ) = cos x+ sin x+ ( xsin x xcos x) [0] 4 4

7 A Numerical Metod for Linear ODEs using Non-recursive Haar 435 Numerical solution for yx ( ) is obtained using te equations (9) (0) and te steps outlined in te algoritm e results are sown in able able : Numerical of Example x Analytical Proposed-metod Absolute Error L norm error in proposed metod (%): % L norm error in literature [0] metod (%): % Example 4 d y( x) Solve te equation + xy( x) = 6sinx+ xsin x, x = [0 ) 4 3 d y d y dy Given initial conditions are ( 0 3 ) = 8, ( 0) = 0, ( 0) = and y ( 0) = 0 Exact solution reported in te literature is yx ( ) = sinx[0] Numerical solution for yx ( ) is obtained using te equations (9) (0) and te steps outlined in te algoritm e results are sown in able

8 436 Monika Garg and Lillie Dewan able : Numerical of Example x Analytical Proposed-metod Absolute Error L norm error in proposed metod (%): 0683 % L norm error in literature [0] metod (%): % Example 3 dy( x) x Solve te equation eyx ( ) x, + = x [0 ) Given initial condition is y ( 0) = 4 Exact solution reported in te literature is 4 3 x yx ( ) = 4 4 x+ x+ (upto first four nonzero terms only) [0] Numerical solution for ( ) yx is obtained using te equations (9) (0) and te steps outlined in te algoritm e results are sown in able 3

9 A Numerical Metod for Linear ODEs using Non-recursive Haar 437 able 3: Numerical of Example 3 x Analytical Proposed-metod Absolute Error L norm error in proposed metod for first four values for comparison purposes (%): % L norm error in literature [0] metod (%): % Example 4 Solve te Scrodinger equation ( α ) Given initial condition is ( ) d φ( x) + me x φ( x) = 0, x = [0 ) φ 0 = and final condition is φ () = e Initial dφ condition on (0) is found using (4) and final condition x Exact solution reported in te literature for α = and me = is φ ( x) = e [] Numerical solution for φ ( x) is obtained using te equations (9) (0) and te steps outlined in te algoritm e results are sown in able 4

10 438 Monika Garg and Lillie Dewan able 4: Numerical of Example 4 x Analytical Proposed-metod Absolute Error L norm error in proposed metod (%): % Results and Discussion Different types of ODEs are solved using te proposed metod and L norm errors are computed It is found tat te errors in te solution by te proposed metod is muc less tan tose reported in te literature A comparison is summarized in able 5 Moreover, Scrodinger equation in Example 4 constitutes te case were complete solution space is to be computed for different values of parameters α and me It as been demonstrated tat computed solutions are in close agreement wit te exact solutions as evident from te L error and since te proposed metod is very computationally efficient so te complete solution space can be easily determined by multiple iterations

11 A Numerical Metod for Linear ODEs using Non-recursive Haar 439 able 5: Comparison of Relative errors Relative error ( L - norm) in % No Example Proposed Metod Reported Metod (Literature)[0] Example Example 3 Example Not available Conclusion Homogeneous and Inomogeneous linear ordinary differential equations wit constant and variable coefficients are solved successfully using non-recursive Haar connection coefficients e caracteristics of te proposed metod of being generalized, computationally efficient and accurate are establised wit te elp of various numerical examples and demonstrated by te computations of L - norm errors in eac case L - norm errors obtained by te proposed metod are muc less as compared to tose reported in te literature e future scope lies in te application of te proposed metod to oter categories of differential equations like functional, delay, partial and fractional differential equations References [] N M Bujurke, C S Salimat and S C Siralasetti (008), Computation of eigenvalues and solutions of regular Sturm-Liouville problems using Haar wavelets, Journal of Computational and Applied Matematics arcive, Vol 9(), pp: 90-0 [] Ülo Lepik (005), Numerical solution of differential equations using Haar wavelets, Matematics and Computers in Simulation Vol 68(), pp: 7-43 [3] Ülo Lepik (008), Haar wavelet metod for solving iger order differential equations, International Journal of Matematics & Computation, Vol (08), pp: [4] Daubecies I (990), e wavelet transform, time-frequency localization and signal analysis, IEEE rans Infor eory, Vol 36, pp:

12 440 Monika Garg and Lillie Dewan [5] Cen, CF and Hsiao CH (997), Haar wavelet metod for solving lumped and distributed-parameter systems IEE Proc Control eory Appl, Vol 44, pp [6] Cun-ui Hsiao (005), Numerical solution of stiff differential equations via Haar wavelets, International Journal of Computer Matematics, Vol 8(9), pp: 7 3 [7] C H Hsiao (00), Wavelets approac to time-varying functional differential equations, International Journal of Computer Matematics, Vol 87(3), pp: [8] Wu JL, Cen CH and Cen CF (000), A unified derivation of operational matrices of integration for integration in system analysis, IEEE Proc Int Conf on Information ecnology: Coding and Computing, pp: [9] Garg M and Dewan L (00), A Novel Metod of Computing Haar Connection Coefficients for Analysis of HCI Systems, Proc of Second nd International Conference on Intelligent Human Computer Interaction, IHCI 00, Lecture Notes in Control and Information Sciences (LNCS), Springer- Verlag, ISBN , pp: [0] Pang Cang, Pang Piau (008), Simple Procedure for te Designation of Haar Wavelet Matrices for Differential Equations, Proceedings of te International Multi Conference of Engineers and Computer Scientists 008, IMECS 008, Hong Kong, Vol II, pp: [] M Saravi, F Asrafi, and SR Mirrajei (009), Numerical of Linear Ordinary Differential Equations in Quantum Cemistry by Clensaw Metod, World Academy of Science, Engineering and ecnology, Issue 49, ISSN: , pp: